Question

$\angle y \cong \angle z$, $\overline{vy} \cong \overline{wz}$, and $\overline{wz} \parallel \overline{vy}$. complete the proof that $\overline{vw} \cong \overline{wx}$.

(image of two triangles with vertices z, x, w and y, v, w, and a table with statements and reasons:

1. $\angle y \cong \angle z$ (given)
2. $\overline{vy} \cong \overline{wz}$ (given)
3. $\overline{wz} \parallel \overline{vy}$ (given)
4. $\angle wvy \cong \angle xwz$ (corresponding angles theorem)
5. blank for statement blank for reason)

Explanation:

Step1: Identify Congruent Triangles

We have $\angle Y \cong \angle Z$, $\overline{VY} \cong \overline{WZ}$, and $\angle WVY \cong \angle XWZ$ (from Corresponding Angles Theorem). So, by ASA (Angle - Side - Angle) congruence criterion, $\triangle VYW \cong \triangle ZWX$.

Step2: Deduce Congruent Sides

Since $\triangle VYW \cong \triangle ZWX$, their corresponding sides are congruent. The corresponding sides for $VW$ and $WX$ are congruent because in $\triangle VYW$ and $\triangle ZWX$, $VW$ and $WX$ are corresponding sides. So, $\overline{VW} \cong \overline{WX}$ as corresponding parts of congruent triangles are congruent (CPCTC). For the statement in step 5, we can say $\triangle VYW \cong \triangle ZWX$ (by ASA) and then use CPCTC to get $\overline{VW} \cong \overline{WX}$. But focusing on the statement part, to prove $\overline{VW} \cong \overline{WX}$, we first establish the triangle congruence. So the statement in step 5 can be $\triangle VYW \cong \triangle ZWX$ with reason ASA (Angle - Side - Angle) because we have two angles ($\angle Y \cong \angle Z$, $\angle WVY \cong \angle XWZ$) and the included side ($\overline{VY} \cong \overline{WZ}$) congruent. Then from triangle congruence, we can conclude $\overline{VW} \cong \overline{WX}$ (CPCTC).

Answer:

The statement for step 5 is $\boldsymbol{\triangle VYW \cong \triangle ZWX}$ with reason "ASA (Angle - Side - Angle) Congruence Criterion" (and then by CPCTC we get $\overline{VW} \cong \overline{WX}$). If we just need the statement to lead to $\overline{VW} \cong \overline{WX}$, the key statement is $\triangle VYW \cong \triangle ZWX$ (by ASA) and then using CPCTC, $\overline{VW} \cong \overline{WX}$. So the statement in step 5 is $\triangle VYW \cong \triangle ZWX$.